Saturday, February 6, 2010

Computational Commutative Algebra with Singular - Ideals, Varieties, and Gröbner Basis

I finally finished my MAT5 (semi-thesis) last week. The exam went just fine even though I was sick I felt nauseous all day. Well, if you feel like learning about of ideals, varieties, and Gröbner bases in a polynomial ring here is my project.

We will begin by introducing some of the basic objects of the reports, such as the polynomial ring, ideals, and varieties. The computer algebra system Singular will be used throughout the report to help with the calculations.
The Gröbner bases will be introduced, and these will allow us solve various problems, such as to determine if a given polynomial lies in a given ideal, and to ease the calculation of common solutions for a set of equations.
The correspondence between ideals and varieties will be proved, and examples will be given to show these correspondences. In the last part, mappings between varieties and the quotient of the polynomial ring modulo an ideal will be studied.

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